Metamath Proof Explorer


Theorem abs2sqlei

Description: The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007)

Ref Expression
Hypotheses abs2sqlei.1 𝐴 ∈ ℂ
abs2sqlei.2 𝐵 ∈ ℂ
Assertion abs2sqlei ( ( abs ‘ 𝐴 ) ≤ ( abs ‘ 𝐵 ) ↔ ( ( abs ‘ 𝐴 ) ↑ 2 ) ≤ ( ( abs ‘ 𝐵 ) ↑ 2 ) )

Proof

Step Hyp Ref Expression
1 abs2sqlei.1 𝐴 ∈ ℂ
2 abs2sqlei.2 𝐵 ∈ ℂ
3 1 absge0i 0 ≤ ( abs ‘ 𝐴 )
4 2 absge0i 0 ≤ ( abs ‘ 𝐵 )
5 1 abscli ( abs ‘ 𝐴 ) ∈ ℝ
6 2 abscli ( abs ‘ 𝐵 ) ∈ ℝ
7 5 6 le2sqi ( ( 0 ≤ ( abs ‘ 𝐴 ) ∧ 0 ≤ ( abs ‘ 𝐵 ) ) → ( ( abs ‘ 𝐴 ) ≤ ( abs ‘ 𝐵 ) ↔ ( ( abs ‘ 𝐴 ) ↑ 2 ) ≤ ( ( abs ‘ 𝐵 ) ↑ 2 ) ) )
8 3 4 7 mp2an ( ( abs ‘ 𝐴 ) ≤ ( abs ‘ 𝐵 ) ↔ ( ( abs ‘ 𝐴 ) ↑ 2 ) ≤ ( ( abs ‘ 𝐵 ) ↑ 2 ) )